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Domain of a logarithm with a variable base

This type of question, asked 4 times: 29 Jan 2023, 29 Jan 2025, 24 Jan 2026 S1, 2 Apr 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    29 Jan 2023

    66
    The domain of f(x)=log⁡(x+1)(x−2)e2log⁡ex−(2x+3),x∈Rf(x) =\frac{\log_{(x + 1)}(x - 2)}{e^{2\log_{e}x}- (2x + 3)},x \in R is

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  2. Question 2

    29 Jan 2025

    129
    If the domain of the function log⁡5(18x−x2−77)\log_{5}\left( 18x -x^{2}- 77 \right) is (α,β)(\alpha,\beta) and the domain of the functionlog⁡(x−1)(2x2+3x−2x2−3x−4)\log_{(x - 1)}\left( \frac{2x^{2}+ 3x - 2}{x^{2}- 3x - 4} \right) is (γ,δ)(\gamma,\delta), then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is equal to :

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  3. Question 3

    24 Jan 2026 S1

    56
    If the domain of the function f(x)=log⁡(10x2−17x+7)(18x2−11x+1)f(x) =\log_{\left( 10x^{2}- 17x + 7 \right)}\left( 18x^{2}- 11x + 1 \right) is (−∞,a)∪(b,c)∪(d,∞)−{e}\mathbf{( - \infty,a) \cup (b,c) \cup (d,\infty) - \{ e\}}, then 90(a+b+c+d+e)90(a + b + c + d + e) equals:

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  4. Question 4

    2 Apr 2026 S1

    71
    If the domain of the function f(x)=log⁡(0.6)(∣2x−5x2−4∣)f(x) = \sqrt{\log_{(0.6)}\left( \left| \frac{2x - 5}{x^{2} - 4} \right| \right)} is (−∞,a]∪{b}∪[c,d)∪(e,∞)( - \infty,a\rbrack \cup \{ b\} \cup \lbrack c,d) \cup (e,\infty), then the value of a+b+c+d+ea + b + c + d + e is ____\_\_\_\_ .